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On the global attractivity and the periodic character of some difference equations. (English) Zbl 0993.39008
Asymptotic properties of solutions of the \(k\)-th order difference equation \[ x_{n+1}=\frac{A_0}{x_n}+\frac{A_1}{x_{n-1}}+\dots+ \frac{A_{k-1}}{x_{n-k+1}},\quad n\in \mathbb N=\{0,1,\dots\} \tag{*} \] are investigated. It is shown that under some restrictions on the numbers \(A_0,\dots,A_{k-1}\) every positive solution of (*) converges to a \(p\)-periodic solution, where the period \(p\) is determined in terms of the coefficients \(A_0,\dots,A_{k-1}\). The main result of the paper reads as follows.
Theorem. Let \(A_0,\dots,A_{k-1}\) be nonnegative real numbers and suppose that the set \(J=\{j\geq 1:\;A_{j-1}>0\}\) is nonempty. Set \(L=\{i+j: i,j\in J\}\), and let \(p=2(\langle L\rangle +1)-\langle L\rangle/\langle J\rangle\), where \(\langle \cdot\rangle\) denotes the greatest common divisor of the elements of the set indicated. Then every positive solution of (*) converges to a periodic solution of (*) with (not necessarily prime) period \(p\). Moreover, there exist solutions of (*) which are periodic with prime period \(p\).

MSC:
39A12 Discrete version of topics in analysis
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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